A TURNPIKE THEOREM FOR A FAMILY OF FUNCTIONS* - Ivie

With these preliminaries in place, we proceed to obtain a turnpike theorem
for the familyU.
3. A turnpike theorem for the familyU
As we restrict the analysis to the setX, henceforth we will assume that all felicity
functions are de…ned onX.
Lemma 2. The family U is (uniformly) equicontinuous, that is: for all " > 0,
there is±>0 such that: jx 1 ¡x 2 j ± implies ju(x 1 ) ¡u(x 2 )j "; for allu2U;
and all(x 1 ;x 2 ) 2X £X.
Proof : Routine and omitted. ¤
Remark 3 We note that a similar result can be proved for the family of the
…rst derivatives, and for the family of the second derivatives, that is, for the
following families:
wherei;j 2 f1;2g.
U i = fD i u:X ! < n j u 2Ug
n
o
U ij = D ij u:X ! < n2 j u 2U ; 6
Henceforth whenever we have fu n g ½U; and eitheru n !uor lim u n =u;
n!1
the limit is taken in the norm of thesup; that is juj=sup ju(x)j; and similarly
x2X
when we deal with sequences in the familiesU i (i 2 f1;2g) andU ij (i;j 2 f1;2g).
6 D ij u(x;y) := D j (D i u)(x;y)
15